3.7. Set of Odd Positive Integers

Try these first

1. 1.

   List three odd positive integers.

   Hint

   Start counting from 1 and only include numbers not divisible by 2.

2. 2.

   Is the set of all odd positive integers finite or infinite?

   Hint

   Consider how the odd positive integers continue indefinitely.

3. 3.

   What defines a countable set?

   • Must be finite
   • Can be matched with positive integers
   • Has no elements

   Hint

   Think about how we define sizes of sets.

4. 4.

   True or False: The set of odd positive integers has more elements than the set of positive integers.

   • True
   • False

   Hint

   Consider the mapping function we introduced.

5. 5.

   Prove that if two sets have the same cardinality, and one of them is finite, then the other must be finite as well.

   Hint

   Consider how bijections relate to the size of both sets.

6. 6.

   Demonstrate a bijection between the set of rational numbers and the set of positive integers, citing your methodology.

   Hint

   Think about how you can systematically traverse all fractions.